LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Regular Sequence Tail On Quotient

Statement

A nonempty sequence $(x_{1}, \dots, x_{n})$ is $M$ -regular if and only if $x_{1}$ is injective on $M$ and $(x_{2}, \dots, x_{n})$ is regular on $M / x_{1} M$ .

Facts & Assumptions

Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Regular Sequence On A Module.

Proof

technique · direct

1.1

Separate the first condition: $x_{1}$ must be injective on $M$ .

givenalgebra

2.1

Every remaining condition is precisely the regularity condition for the tail on $M / x_{1} M$ , including the nonzero terminal quotient.

step 1.1algebra∎

Depends on

Regular Sequence On A Module

Used by

Initial Subsequences Of A Regular Sequence Are Regular Corollary

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Koszul complexes and regular sequences (standard reference, not scraped)